This is a profound result, independent of the solution of the Cauchy problem for the Boltzmann flow posed in Theorem REF . It provides the basic component not only a coercive condition that yields global stability of solutions but also the control of high energy tails or exponential moments to be discussed in Subsection  .
It has its roots in the work of Povzner [1]}, Desvillettes [2]} and Wennberg[3]}, but it was the ground breaking work of Bobylev [4]} who introduced, by means of the energy identity (REF ), the concept of angular averaging that produces a broad and natural argument to control the integral of the sum of convex weight functions evaluated in post-collisional velocities proportional to the angular transition function \({\bf b}(\hat{u}\cdot \sigma ) \in L^1(\mathbb {S}^{d-1} )\) , without any need to omit point of singularities. While the original work in [4]} was developed for the case of hard potentials an three dimensions (i.e. \(\gamma =1\)  and \({\bf b}(\hat{u}\cdot \sigma )\)  being constant), we
present in this review
ideas techniques that are slight modifications of those from [6]}, [7]}, [8]}. These new angular averaging approaches have been recently developed for a system of multi-component gas mixtures [9]}, and more recently for a broad class of cross sections and transition probabilities for the case of polyatomic gases [10]}.
